Analysis of Function Component Complexity for Hypercube Homotopy Algorithms

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that globally convergent from an arbitrary starting point with probability one. The essence of these homotopy algorithms is the construction of a homotopy map p-sub a and the subsequent tracking of a smooth curve y in the zero set p-sub a to the -1 (0) of p-sub a. Tracking the zero curve y requires repeated evaluation of the map p-sub a, its n x (v + 1) Jacobian matrix Dp-sub a and numerical linear algebra for calculating the kernel of Dp-sub a. This paper analyzes parallel homotopy algorithms on a hypercube, considering the numerical algebra, several communications topologies and problem decomposition strategies, functions component complexity, problem size, and the effect of different component complexity distributions. These parameters interact in complicated ways, but some general principles can be inferred based on empirical results

Unit Tangent Vector Computation for Homotopy Curve Tracking on aHypercube

Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map and subsequent tracking of some smooth curve in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vectors at different points along the zero curve. Because of the way a homotopy map is constructed, the unit tangent vector at each point in the zero curve of a homotopy map (symbols) is in the kernel of the Jacobian matrix (symbols). Hence tracking the zero curve of a homotopy map involves finding the kernel of the Jacobian matrix (symbols). The Jacobian matrix (symbols) is a n x (n + 1) matrix with full rank. Since the accuracy of the unit tangent vector is very important, on orthogonal factorization instead of an LU factorization of the Jacobian matrix is computed. Two related factorizations, namely QR and LQ factorization, are considered here. This paper presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms on a hypercube. Since the purpose of this study is to find ways to parallelize homotopy algorithms, it is assumed that the matrices are small, dense, and have a special structure such as that of the Jacobian matrix of a homotopy map

Efficacy of Stochastic Vestibular Stimulation to Improve Locomotor Performance in a Discordant Sensory Environment

Astronauts exposed to microgravity face sensorimotor challenges incurred when readapting to a gravitational environment. Sensorimotor Adaptability (SA) training has been proposed as a countermeasure to improve locomotor performance during re-adaptation, and it is suggested that the benefits of SA training may be further enhanced by improving detection of weak sensory signals via mechanisms such as stochastic resonance when a non-zero level of stochastic white noise based electrical stimulation is applied to the vestibular system (stochastic vestibular stimulation, SVS). The purpose of this study was to test the efficacy of using SVS to improve short-term adaptation in a sensory discordant environment during performance of a locomotor task

Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube

Polynomial systems of equations frequently arise in many applications such as solid modeling, robotics, computer vision, chemistry, chemical engineering, and mechanical engineering . Locally convergent iterative methods such as quasi-Newton methods may diverge or fail to find all meaningful solutions of a polynomial system. Recently a homotopy algorithm has been proposed for polynomial systems that is guaranteed globally convergent (always converges from an arbitrary starting point) with probability one, finds all solutions to the polynomial system, and has a large amount of inherent parallelism. There are several ways the homotopy algorithms can be decomposed to run on a hypercube. The granularity of a decomposition has a profound effect on the performance of the algorithm. The results of decompositions with two different granularities are presented. The experiments were conducted on an iPSC-16 hypercube using actual industrial problems

Invasive frogs show persistent physiological differences to elevation and acclimate to colder temperatures

The coqui frog (Eleutherodactylus coqui) was introduced to the island of Hawai’i in the 1980s and has spread across much of the island. Concern remains that this frog will continue to expand its range and invade higher elevation habitats where much of the island’s endemic species are found. We determined whether coqui thermal tolerance and physiology change along Hawai’i’s elevational gradients. We measured physiological responses using a short-term experiment to determine baseline tolerance and physiology by elevation, and a long-term experiment to determine the coqui’s ability to acclimate to different temperatures. We collected frogs from low, medium, and high elevations. After both the short and long-term experiments, we measured critical thermal minimum (CTmin), blood glucose, oxidative stress, and corticosterone levels. CTmin was lower in high elevation frogs than low elevation frogs after the short acclimation experiment, signifying that they acclimate to local conditions. After the extended acclimation, CTmin was lower in frogs acclimated to cold temperatures compared to warm-acclimated frogs and no longer varied by elevation. Blood glucose levels were positively correlated with elevation even after the extended acclimation, suggesting glucose may also be related to lower temperatures. Oxidative stress was higher in females than males, and corticosterone was not significantly related to any predictor variables. The extended acclimation experiment showed that coquis can adjust their thermal tolerance to different temperatures over a 3-week period, suggesting the expansion of coqui into higher elevation habitats may still be possible, and they may not be as restricted by cold temperatures as previously thought

High Dimensional Homotopy Curve Tracking on a Shared Memory Multiprocessor

Results are reported for a series of experiments involving numerical curve tracking on a shared memory parallel computer. Several algorithms exist for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then the tracking of some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. A parallel version of HOMPACK is implemented on a shared memory parallel computer with various levels and degrees of parallelism (e.g., linear algebra, function and Jacobian matrix evaluation), and a detailed study is presented for each of these levels with respect to the speedup in execution time obtained with the parallelism, the time spent implementing the parallel code and the extra memory allocated by the parallel algorithm